1. 16722 mws@cmu.edu Mo:20090201synchronous detection93+1 assignment (15)

2. 16722 mws@cmu.edu Mo:20090201synchronous detection93+2 assignment (16)

3. 16722 mws@cmu.edu We:20090203measurements & distributions107+3 next topic... errors and uncertainties in measurement combining errors/uncertainties to find the net error/uncertainty in a quantity calculated from several component measurements, assuming you know the individual error/uncertainty in each component (error propagation) to find the net error/uncertainty you can realize if you make and combine several measurements of the same quantity, perhaps using different sensors, instruments, or sensing principles (sensor fusion)

4. 16722 mws@cmu.edu We:20090203measurements & distributions107+4 key points to take away... accuracy & precision of single measurements combining errors/uncertainties a result calculated from multiple different uncertain measurements must have a relative uncertainty worse than the worst component’s relative uncertainty a result calculated from multiple uncertain measurements of the same measurand, e.g., by making the measurement several times with different instruments, must have an absolute uncertainty better than the best individual measurement’s absolute uncertainty

5. 16722 mws@cmu.edu We:20090203measurements & distributions107+5 accuracy & precision accuracy: how close is your measurement to the correct value? (whatever “correct” means...) precision: what is the spread of your measurements around the value you found?

6. 16722 mws@cmu.edu We:20090203measurements & distributions107+6 error & uncertainty the difference between error & uncertainty is... well... that’s actually a little uncertain... generally “uncertainty” is used more-or-less interchangeably with the resolution of the measuring instrument... and error is used more-or-less interchangeably with the inaccuracy of the result errors you can characterize by the statistical distribution of the measurements you actually make errors you can only estimate, i.e., guess fundamental arguments among mathematicians etc about whether the same arithmetic applies to both

7. 16722 mws@cmu.edu We:20090203measurements & distributions107+7 distribution of measurements whose outcome is an integer count

8. 16722 mws@cmu.edu We:20090203measurements & distributions107+8 measurements whose outcomes are integer counts observed number of counts relative probability (1) you count many samples (2) you calculate the mean of the counts in your samples (3) what determines the width of this distribution of observed results?

9. 16722 mws@cmu.edu We:20090203measurements & distributions107+9 assignment (20) The World Series problem: a) Get ~100 years of outcomes of the World Series; http://www.baseball- almanac.com/ws/wsmenu.shtml is one of many compilations on the web.http://www.baseball- almanac.com/ws/wsmenu.shtml b) If the teams are equally matched then the probability of the series going 4, 5, 6, or 7 games is equivalent to coin flipping; plot actual outcomes vs. corresponding coin-flip probabilities. c) Now compare the probabilities you calculated with the actual data. Does the series generally run shorter or longer than you would expect? What are some reasons actual and expected might differ?

10. 16722 mws@cmu.edu Mo:20090201synchronous detection93+10 assignment (17) more good words to think about: integrator ↔ averager or smoother differentiator ↔ edge or transition detector

11. 16722 mws@cmu.edu Mo:20090201synchronous detection93+11 summary, then details narrow banding “squeezes out” noise but not signal – problem: to keep the system “on station” lock-in amplifier (a.k.a. synchronous amplifier, synchronous rectifier, phase sensitive detector, etc) – stays “on station” because you have the reference signal – squeezes out noise in phase and frequency domains generalization: some forms of modern spread spectrum communication coding, e.g., CDMA (code division multiple access) for GPS, cell phones, etc.

12. 16722 mws@cmu.edu Mo:20090201synchronous detection93+12 the concept expressed as code ordinary passive “full-wave rectifier”: – Abs[signal(t)] = signal(t)>0 ? signal(t):- signal(t) synchronous rectifier: – SyncAbs[signal(t), reference(t)] = reference(t)>0 ? signal(t):-signal(t) – reference(t) takes on many application- and economics-dependent forms: multiply by cos(ω ref t): typical analog modulation on/off: typical mechanical or digital modulation pseudo-random word: typical broadband system – CDMA cell phone communication – GPS satellite navigation system

13. 16722 mws@cmu.edu Mo:20090201synchronous detection93+13 graphical illustration (1) consider an illustrative “slowly varying” signal S[t] = 1.0 - 0.8t + 0.8t 2 + 0.2 sin[4  t]+0.025 cos[12  t]

14. 16722 mws@cmu.edu Mo:20090201synchronous detection93+14 (2) now bury the signal in a much larger sea of noise: noise[t] = 2.+.4t-4.t 2 -.5Sin[4  t+  /6.] -.25Cos[12  t-  /12.] signal+noise signal

15. 16722 mws@cmu.edu Mo:20090201synchronous detection93+15 (3) when the signal (but not the noise) is modulated this is how the synchronous rectifier output looks: notice that the average value (the solid line) faithfully reconstructs the original signal … now we need to actually implement the averaging what you see what you would see if you ran this through an averaging algorithm (S+N) above (N) below

16. 16722 mws@cmu.edu Mo:20090201synchronous detection93+16 (4) here it is after processing by a 21-point sliding average: heavier filtering would faithfully restore the original signal, distorted (under favorable conditions only slightly) by the noise that happens to be in the frequency and phase band accepted by the process

17. 16722 mws@cmu.edu Mo:20090201synchronous detection93+17 summary … if you can give your signal (but not the noise) some sort of “signature” you can pull it back out it of a lot of noise if you chop or sinusoidally modulate it and use a detection method that looks for the modulation frequency you win nicely if you use a detection method that looks for the modulation frequency and its phase then you win spectacularly more complex signatures, e.g., CDMA, provide even better noise rejection

18. 16722 mws@cmu.edu Mo:20090201synchronous detection93+18 assignment 18) The three major cell phone systems are CDMA, TDMA, and GSM. xDMA describes both the system and the coding method; GSM describes a system that uses a kind of “phase shift coding” (PSC). Identify (e.g., by web search) and document one service provider who uses each coding method. Briefly describe the essential idea underlying each coding method. [Note: you may find that they are not mutually exclusive, e.g., some systems use both PSC and TDMA. ]

19. 16722 mws@cmu.edu Mo:20090201synchronous detection93+19 last few topics we covered... basic transduction & measurement sensors that deliver electronic signals – voltage sources & current sources – device parameters (resistance, capacitance, …) force a voltage, measure the current force a current, measure the voltage mother nature’s efforts to thwart sensing – fundamental noise (thermal, shot, “flicker”) – technical noise (the environment) some tricks to make the best of it – narrow banding in frequency domain – coding to narrow-band in phase domain

20. 16722 mws@cmu.edu Mo:20090201synchronous detection93+20 usually signal accumulates faster than noise – e.g., integrated signal power ~ measuring time integrated noise power ~ measuring time ½ so S:N ratio gets better as time ½ – but not always! e.g., in an inertial navigation system signal-to-noise ratio of a position measurement gets worse as time 3/2 remember well...

21. 16722 mws@cmu.edu Mo:20090201synchronous detection93+21 assignment 19) Numerically simulate a one-dimensional “integrated random walk”: you are in a sealed-cabin with an accelerometer; you know your starting position, and you start with zero velocity; integrating the accelerometer output gives your velocity, integrating again gives your displacement from initial position; but the accelerometer output has random noise on it. Show that your simulation suggests Δv ~ t 1/2 and Δx ~ t 3/2.